Proximity two bands Eliashberg theory of electrostatic field-effect doping in a superconducting film of MgB.

A key aspect of field effect experiments is the possibility to induce charges on the first layers of a sample as a function of an applied gate voltage. It is therefore possible to study correlated phases of matter as a function of the induced charge density and the applied electric field. Moreover, resulting charge modulation along the direction of the applied electric field gives rise to junctions between perturbed and uneffected regions of the sample. In the framework of proximity effect Eliashberg theory, we investigate the consequence of an applied static electric field on the transition temperature of a two-band s-wave superconductor magnesium diboride. In most cases the only free parameter in the theory is the penetration depth of the applied electric field, whereas there is no freedom when the static perturbation is sufficiently weak. We come to the conclusion that the optimal way to enhance the critical temperature is to have a very thin film of magnesium diboride, otherwise the external electric field would not have substantial effect on superconductivity in this material.